Interesting. So c² would also equal 1. Which (in those scaled units) implies E = m.
This not only greatly simplifies the relationship but actually makes much more sense that way.
Not only does this clarify the relationships, but in many ways, these "natural constants" are an artifact from our past ignorance. Boltzmann's constant, in a way, is nothing more than a "conversion factor" that we hold over from the time when we believed temperature and energy to be two separate concepts. In the same way, the speed of light is an artifact from a time when we considered time and space to be distinct concepts, measuring them in distinct units, and needing a conversion factor (in units of distance/time) to map between them.
It's as if we would all collectively agree that the "up" direction, from now on, would be measured in floops, and the slope of a hill would be measured in floops/meter.
From a philosophical point of view, it's not just saying "measure time in units such that c = 1". It's saying "let's consider time to be a distance, and measure it in the same units as we do the other ones".
* https://www.youtube.com/watch?v=ZbGxXyqlhbU (FloatHeadPhysics on this)
One physics convention just sets its value to 1. All of those Minkowsky diagrams that we see are measured in light seconds on the space axis, in order to make c have the value of 1 space unit per time unit; so all of the graphical sheep, spaceships, cats, people, torches and stuff that are placed upon them are very much not to scale. (-: